Session N40: Quantum Algorithms for Computational Chemistry

Numerical Simulations of Noisy Quantum Circuits for Computational Chemistry

Opportunities afforded by near-term quantum computers to simulate the properties of small molecules depend on the structure of the computational ansatz as well as the errors induced by device noise. Here we examine the influence of these factors on the widely used variational quantum eigensolver (VQE) algorithm for quantum chemistry by investigating the behavior of noisy ansatz circuits with numerical simulation. Using ansatz circuits derived from unitary coupled cluster (UCC) theory, we estimate the ground-state electronic energy of few-electron models to show how relative error and fidelity scale with levels of gate-based noise, inter-molecular configuration, ansatz depth, and optimization methods. We find ansatz depth is the leading indicator for fidelity and relative error, while error itself is robust to the level of theory and optimization method.    

Fast-forwarding quantum evolution

We describe the problem of fast-forwarding quantum evolution, whereby the dynamics of certain quantum systems can be simulated with gate complexity that is sublinear in the evolution time, and present some examples where fast-forwarding is attained. In particular, we show that some local spin systems whose Hamiltonians can be taken into block diagonal form using an efficient quantum circuit, such as those that are permutation-invariant, can be exponentially fast-forwarded. We also show that certain classes of positive semidefinite local spin systems, also known as frustration-free, can be polynomially fast-forwarded, provided the initial state is supported on a subspace of sufficiently low energies. Last, we show that all quadratic fermionic systems and number-conserving quadratic bosonic systems can be exponentially fast-forwarded in a model wherequantum gates are exponentials of specific fermionic or bosonic operators, respectively. Our results extend the classes of physical Hamiltonians that were previously known to be fast-forwarded, while not necessarily requiring methods that diagonalize the Hamiltonians efficiently. We further develop a connection between fast-forwarding and precise energy measurements that also accounts for polynomial improvements.   

Fermion-to-Qubit Encodings for Quantum Simulation

Quantum simulation of fermionic many-body systems is an important application of quantum computing, with relevance to problems in the fields of quantum chemistry and condensed matter. One challenge in implementing these simulations is that they require mapping the indistinguishable fermions to distinguishable qubits. Several schemes for accomplishing this fermion to qubit mapping have been proposed. In this presentation we provide an overview of progress in the field. We will highlight a number of fermionic encodings, ranging from the well-known Jordan-Wigner mapping to the recent Custom Fermionic Codes scheme developed by R. Chien and J. Whitfield (arXiv:2009.11860).   

Adaptive Variational Quantum Imaginary Time Evolution Approach for Ground State Preparation

An adaptive variational quantum imaginary time evolution (AVQITE) approach is introduced that yields efficient representations of ground states for interacting Hamiltonians on near-term quantum computers. It is based on McLachlan's variational principle applied to the imaginary time evolution of variational wave functions. The variational parameters evolve deterministically according to equations of motion that minimize the difference to the exact imaginary time evolution, which is quantified by the McLachlan distance. Rather than working with a fixed variational ansatz, where the McLachlan distance is constrained by the quality of the ansatz, the AVQITE method iteratively expands the ansatz along the dynamical path to keep the McLachlan distance below a chosen threshold. This ensures that the state is able to follow the quantum imaginary time evolution path in the system Hilbert space rather than in a restricted variational manifold set by a predefined fixed ansatz. AVQITE is used to prepare ground states of H$_4$, H$_2$O and BeH$_2$ molecules, where it yields compact variational ans\"atze and ground state energies within chemical accuracy. Polynomial scaling of circuit depth with system size is shown through a set of AVQITE calculations of quantum spin models. Finally, quantum Lanczos calculations are demonstrated alongside AVQITE without additional quantum resource costs.   

Measurement-Based Time Evolution for Quantum Simulation of Fermionic Systems

Quantum simulation within the quantum phase estimation-based family of algorithms yields exact eigenenergies for fermionic models that are otherwise intractable due to the Monte Carlo sign problem. In circuit-based quantum computation (CBQC), such algorithms repeatedly apply quantum gates to an input wavefunction to achieve time evolution for runtimes that increase exponentially with required bit precision. In measurement-based quantum computation (MBQC), time evolution is effectively driven by sequences of local measurements on an initial entangled resource state. We propose that the gate time burden in CBQC quantum simulation can be shifted to a burden on measurement precision in MBQC quantum simulation. Along these lines, we construct example single-qubit measurement patterns to implement MBQC algorithms for Kitaev and Hubbard chains. We also construct an example hybrid MBQC algorithm with a subroutine for eigenvalue estimation using an offline time series. We consider the scaling of measurements, precision costs, and tolerable errors in the MBQC algorithm. Our examples show that MBQC eigenvalue estimation yields a runtime advantage over CBQC when measurements can be performed quickly and accurately.   

Quantum Algorithms for Ground State Preparation and Green's Function Calculation

We propose quantum algorithms for projective ground-state preparation and frequency-domain Green's function calculations. The algorithms are based on the linear combination of unitary (LCU) framework and use only quantum resources. To prepare the ground state, we apply the operator exp(-τH²) expressed using LCU on an easy-to-prepare initial state. This procedure saturates the near-optimal scaling O(log(1/(γη))/(γΔ)) of other algorithms, in terms of the spectral gap ∆, the targeted error η, and the overlap γ between the initial state and the exact ground state. Our algorithm can easily be combined with the spectral gap amplification technique to achieve better scaling O(1/√Δ) for frustration-free Hamiltonians. To compute single and multi-particle response functions, we act on the prepared ground state with the retarded resolvent operator in the LCU form derived from the Fourier-Laplace integral transform (FIT). Our resolvent algorithm has the complexity O(log(1/(Γϵ))/Γ²) for the frequency resolution Γ of the response functions and the targeted error ϵ, while classical algorithms for FIT usually have polynomial scaling over the error ϵ. To illustrate the complexity scaling of our algorithms, we provide numerical results for their application to the paradigmatic Fermi-Hubbard model.   

Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer

Many-electron problems pose some of the greatest challenges in computational science, with important applications across many fields of modern science. Fermionic quantum Monte Carlo (QMC) methods are among the most powerful approaches to these problems. However, they can be severely biased when controlling the fermionic sign problem using constraints, as is necessary for scalability. Here we propose an approach that combines constrained QMC with quantum computation to reduce such biases. We experimentally implement our scheme using up to 16 qubits in order to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed on quantum computers (more than doubling the size of prior electron correlation calculations), while obtaining accuracy competitive with state-of-the-art classical methods. Our results demonstrate a new paradigm of hybrid quantum-classical algorithm, surpassing the popular variational quantum eigensolver in terms of potential towards the first practical quantum advantage in ground state many-electron calculations. Compared with the variational quantum eigensolver, our new hybrid quantum-classical computational paradigm offers an alternative path towards achieving a practical quantum advantage for the electronic structure problem without demanding exceedingly accurate preparation and measurement of the exact ground state wavefunction.   

Holographic simulation of correlated electrons and thermal states on a trapped-ion quantum processor

We present holographic quantum simulation algorithms to prepare ground- and thermal-states of interacting electron materials as quantum matrix product states (qMPS) or density operators (qMPDO) respectively, requiring far-fewer qubits than the number of orbitals being simulated. We demonstrate both algorithms through numerical simulations and experimental implementations on Honeywell’s H1 trapped-ion quantum processor. First, we adapt a method to compress mean-field fermion states into qMPS form, achieving polynomial reduction of qubit and gate resources compared to recent quantum Hartree-Fock demonstrations, making them ideal for near-term implementations. We show that such mean-field qMPS serve as a useful starting point for variational refinement to build in correlations, and argue that it enables preparation of qMPS with exponentially large bond dimension using polynomial circuit resources. Second, we devise a method for implementing thermal state qMPDO by stochastically sampling over an ensemble of qMPS, which can be optimized variationally. We explore the representational power of thermal state qMPDOs for correlated spin-chains and show that they enable simulation of non-equilibrium thermal transport dynamics from initial states with inhomogeneous temperature.   

Insights from an adaptive variational wave function study of the Fermi-Hubbard Model

Approximating the ground states of strongly interacting electron systems in quantum chemistry and condensed matter physics is expected to be one of the earliest applications of quantum computers. In this paper, we prepare highly accurate ground states of the Fermi-Hubbard model for small grids up to 6 sites (12 qubits) by using an interpretable, adaptive variational quantum eigensolver(VQE) called ADAPT-VQE. In contrast with non-adaptive VQE, this algorithm builds a system-specific ansatz by adding an optimal gate built from one-body or two-body fermionic operators at each step. We show this adaptive method outperforms the non-adaptive counterpart in terms of fewer variational parameters, short gate depth, and scaling with the system size. The fidelity and energy of the prepared state appear to improve asymptotically with ansatz depth. We also demonstrate the application of adaptive variational methods by preparing excited states and Green functions using a proposed ADAPT-SSVQE algorithm. Lower depth, asymptotic convergence, noise tolerance of a variational approach, and a highly controllable, system-specific ansatz make the adaptive variational methods particularly well-suited for NISQ devices.   

A Near-Term Quantum Algorithm for Computing Molecular and Materials Properties based on Recursive Variational Series Methods

Determining properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We propose a quantum algorithm to estimate properties of molecules using near-term quantum devices. The method is a recursive variational series estimation method where we expand an operator of interest in terms of Chebyshev polynomials, and evaluate each term in the expansion using a variational quantum algorithm. We test our method on filter operators: the spectral density operator (SDO) $\delta(E-\hat{H})$ and Green's filter operator (GFO) $(E-\hat{H})^{-1}$, where the poles correspond to eigenvalues of the system Hamiltonian. A useful property of the SDO and GFO is to ``filter'' out eigenenergies and eigenstates in the spectral regions of interest, for example to analyze optical spectra. Furthermore, the GFO is useful to study for example quantum many-body systems and quantum transport by calculating the Green's functions in the frequency domain. We discuss two implementations based on our recursive variational series estimation method and a recent study of quantum algorithms to compute Green's functions using the Lanczos recursions [F. Jamet \emph{et al.}, arXiv:2105.13298 (2021)]. Numerically, we show that in the present of sampling and device noise, our method is significantly more noise resilient. In conclusion, we find there is a potentially useful application on near-term quantum computers for evaluating expectation values.   

Unraveling correlated materials' properties with noisy quantum computers: Natural-orbitalized variational quantum eigensolving of extended impurity models within a slave-boson approach

We propose a method for computing space-resolved correlation properties of the two-dimensional Hubbard model within a quantum-classical embedding strategy that uses a Noisy, Intermediate Scale Quantum (NISQ) computer to solve the embedded model. While previous approaches were limited to purely local, one-impurity embedded models, requiring at most four qubits and relatively shallow circuits, we solve a two-impurity model requiring eight qubits with an advanced hybrid scheme on top of the Variational Quantum Eigensolver algorithm. This iterative scheme, dubbed Natural Orbitalization (NOization), gradually transforms the single-particle basis to the approximate Natural-Orbital basis, in which the ground state can be minimally expressed, at the cost of measuring the one-particle reduced density-matrix of the embedded problem. We show that this transformation tends to make the variational optimization of existing (but too deep) ansatz circuits faster and more accurate, and we propose an ansatz, the Multi-Reference Excitation Preserving (MREP) ansatz, that achieves great expressivity without requiring a prohibitive gate count. The one-impurity version of the ansatz has only one parameter, making the ground state preparation a trivial step, which supports the optimal character of our approach. Within a Rotationally Invariant Slave Boson embedding scheme that requires a minimal number of bath sites and does not require computing the full Green's function, the NOization combined with the MREP ansatz allow us to compute reasonably accurate, space-resolved quasiparticle weights and static self-energies for the Hubbard model even in the presence of noise levels representative of current NISQ processors. This paves the way to a controlled solution of the Hubbard model with larger and larger embedded problems solved by quantum computers.   

T-Fermion: A non-Clifford gate cost assessment library of quantum phase estimation algorithms for quantum chemistry

Quantum Phase Estimation is one of the most useful quantum computing algorithms for quantum chemistry and as such, significant effort has been devoted to designing efficient implementations. In this article, we introduce T-Fermion, a library designed to estimate the T-gate cost of such algorithms, for an arbitrary molecule. As examples of usage, we estimate the T-gate cost of a number of simple molecules and compare the same Taylorization algorithms using Gaussian and plane-wave basis. We find out that naïve Taylorization techniques can be surprisingly efficient when used in plane waves and combined with QROM techniques.